International Journal of Mathematics

Fujita’s second theorem for Kähler fibre spaces over a curve asserts, that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $V=A\oplus Q$, where $A$ is ample and $Q$ is unitary flat. We focus on our negative answer [F. Catanese and M. Dettweiler, Answer to a question by Fujita on variation of Hodge structures, to appear in Adv. Stud. Pure Math.] to a question by Fujita: is $V$ semiample? We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group ${(\mathbb{Z}/n)}^2$ of a Del Pezzo surface $Z$ of degree 5 (branched on the union of the lines of $Z$, which form a bianticanonical divisor), and endowed with a semistable fibration with only three singular fibres. The simplest such surfaces are the three ball quotients considered in [I. C. Bauer and F. Catanese, A volume maximizing canonical surface in 3-space, Comment. Math. Helv. 83(1) (2008) 387–406.], fibred over a curve of genus 2, and with fibres of genus 4. These examples are a larger class than the ones corresponding to Shimura curves in the moduli space of Abelian varieties.

Let $\mathrm{\text{Gr}}(2,E)$ be the Grassmann bundle of two-planes associated to a general bundle $E$ over a curve $X$. We prove that an embedding of $\mathrm{\text{Gr}}(2,E)$ by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over $X$, analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over $X$, contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

In this paper, we investigate the relationship between twisted and untwisted character varieties, via a specific instance of the Cohomological Hall algebra for moduli of objects in 3-Calabi–Yau categories introduced by Kontsevich and Soibelman. In terms of Donaldson–Thomas theory, this relationship is completely understood via the calculations of Hausel and Villegas of the $E$ polynomials of twisted character varieties and untwisted character stacks. We present a conjectural lift of this relationship to the cohomological Hall algebra setting.

This paper discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper [Noncommutative deformation and flops, Duke Math. J. 165(8) (2016) 1397–1414], the author and Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It characterizes their geometric and homological properties in a unified manner, using the theory of noncommutative deformations. Toda has now clarified the enumerative significance of the contraction algebra for flopping curves, calculating its dimension in terms of Gopakumar-Vafa invariants [Noncommutative width and Gopakumar–Vafa invariants, Manuscripta Math. 148(3–4) (2015) 521–533]. Before reviewing these results, and others, the author gives a brief introduction to the rich geometry of flopping curves on 3-folds, starting from the resolutions of Kleinian surface singularities. This is based on a talk given at VBAC 2014 in Berlin.

The Joyce integrable system and the corresponding Bridgeland–Toledano-Laredo connections are fundamental objects associated with suitable abelian categories or, more generally, with a class of continuous families of stability data. We offer an overview of some of our work, mostly joint with M. Garcia Fernandez, focusing on equations of TBA type as a useful tool in the analysis of these objects and their deformations, and as a means to establish a connection with tropical geometry.

We prove that the singular locus of a rank $2$ non-locally free instanton sheaf $E$ on ${\mathbb{P}}^3$ has pure dimension $1$. Moreover, we also show that the dual and double dual of $E$ are isomorphic locally free instanton sheaves, and that the sheaves ${ℰ}{\mathrm{\text{xt}}}^1(E,{{𝒪}}_{{\mathbb{P}}^3})$ and $E^{\vee \vee }/E$ are rank $0$ instantons. We also provide explicit examples of instanton sheaves of ranks $3$ and $4$ illustrating that all of these claims are false for higher rank instanton sheaves.

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson–Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker–Maruyama moduli spaces with respect to two different chosen polarizations are related via Thaddeus-flips through other “multi-Gieseker”-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson–Uhlenbeck moduli space.

We discuss the problem of determining the de Rham, Dolbeault and algebraic cohomological dimension of ${{ℳ}}_g$, focusing on possible strategies of attack and then concentrating on exhaustion functions. In the final section, we explain how these techniques can be employed to provide a nontrivial upper bound for the Dolbeault cohomological dimension of ${{ℳ}}_g$.

Let ${{ℳ}}^{\mathrm{\text{st}}}$ (${{ℳ}}^{\mathrm{\text{pst}}}$) be a moduli space of stable (polystable) bundles with fixed determinant on a complex surface with $b_1=1,p_g=0$ and let $Z\subset {{ℳ}}^{\mathrm{\text{st}}}$ be a pure $k$-dimensional analytic set. We prove a general formula for the homological boundary $\delta {[Z]}^{BM}\in H_{2k-1}^{BM}(\partial {\widehat{{ℳ}}}^{\mathrm{\text{pst}}},\mathbb{Z})$ of the Borel–Moore fundamental class of $Z$ in the boundary of the blown up moduli space ${\widehat{{ℳ}}}^{\mathrm{\text{pst}}}$. The proof is based on the holomorphic model theorem of [A. Teleman, Instanton moduli spaces on non-Kählerian surfaces, Holomorphic models around the reduction loci, J. Geom. Phys. 91 (2015) 66–87] which identifies a neighborhood of a boundary component of ${\widehat{{ℳ}}}^{\mathrm{\text{pst}}}$ with a neighborhood of the boundary of a “blown up flip passage”. We then focus on a particular instanton moduli space which intervenes in our program for proving the existence of curves on class VII surfaces. Using our result, combined with general properties of the Donaldson cohomology classes, we prove incidence relations between the Zariski closures (in the considered moduli space) of certain families of extensions. These incidence relations are crucial for understanding the geometry of the moduli space, and cannot be obtained using classical complex geometric deformation theory.

We use the Galois action on ${\pi }_1^{\mathrm{\text{ét}}}({\mathbb{P}}_{\overline{Q}}^1-\{0,1,\infty \})$ to show that the homotopy equivalence $S^1\wedge ({𝔾}_{m,\mathbb{Q}}\vee {𝔾}_{m,\mathbb{Q}})\cong S^1\wedge ({\mathbb{P}}_{\mathbb{Q}}^1-\{0,1,\infty \})$ coming from purity, does not desuspend to a map ${𝔾}_{m,\mathbb{Q}}\vee {𝔾}_{m,\mathbb{Q}}\to {\mathbb{P}}_{\mathbb{Q}}^1-\{0,1,\infty \}$.